Calculating P&L and Delta for a Delta-Hedged Call Spread
Summary
The document asks how to reproduce the profit-and-loss and delta figures for a position involving a long call, a short call at a higher strike, and a forward hedge. The position is European, uses options on the same underlying and maturity, and assumes a flat yield curve. The question notes a possible reversal error in the source example and shows why applying the initial deltas linearly to an underlying price move does not match the reported P&L.
The answer explains that a delta hedge is only locally neutral: as the underlying moves, each option’s delta changes. It illustrates recalculating call deltas at a new underlying price with a Black-Scholes framework, then combining them with the forward’s unit delta to find the position’s new net delta. To obtain P&L across prices, option values should also be recalculated in Black-Scholes. The explanation assumes a volatility input inferred from the given deltas and offers a calculation outline rather than a complete table; the source’s possible position reversal remains a caveat.
Key ideas
- A delta hedge is locally neutral, but the position's net delta changes as the underlying price moves.
- Recalculate each option's delta at the new price using an option-pricing model.
- Add the forward hedge's delta to the option deltas to obtain the portfolio's net delta.
- Calculate P&L from repriced options rather than extending initial deltas linearly across price moves.
- The example depends on an assumed volatility and includes uncertainty about the original position direction.
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Full text
# P/L table for a delta hedged position # P/L table for a delta hedged position I am trying to replicate the table at pag. 119 of Dynamic Hedging by N. Taleb with no success. In the example called "A misleading delta" an operator has the following position: - long \$1 million of the 96 call(delta .824, so \$824k long) - short \$1 million of the 104 call (delta .198, so \$198k short) - hedge the position by selling \$626k of the forward where all options are European on the same underlying asset and with the same maturity. Plus, we assume a flat yield curve. I found in this answer (Question on an example from "Dynamic Hedging" by Nassim Taleb) the fact that, probably, there is an error in the book and that the positions should be reversed. Anyway, I am not able to calculate the P/Ls and deltas reported and I would like some help. For example, from what I understand, the market is trading at \$100 the underlying so we have P/L=0 and delta=0. Suppose that the underlying is at $103. I would compute the P/L by summing (I consider here the "reversed position" as suggested in the answer above): - gain from long the asset: \$626k*(103-100)/100 - gain from long the 104 call: $1million*(103-100)*0,198 - loss from short the 96 call: -\$1million*(103-100)*0,824 What I obtained is very different from the P/L of \$12k reported. How is this last number calculated? How the delta of \$106k is calculated? Thanks for the help. Please, let me know if more details are needed. ## Answer by MrLCh (score 1) https://quant.stackexchange.com/a/79376 Given the delta for the two options I would assume that the volatility is assumed to be about 0.16. If you (delta) hedge your position you will have a P&L close to 0 (as your change in PnL can be approximated by your net delta, which is 0), but as you move further away from the asset price 100 your net delta will change. For example if your price goes to 105 the following applies (assuming 0.16 volatility): delta of the $96$ call: $0.978$ (calculated in Black-Scholes framework) delta of the 104 call: $0.593$ (calculated in Black-Scholes framework) As you keep the the forward you still have a delta 1 position of \$ $626,000$. So your net delta becomes $0.593-0.978+0.626 = 0.241$. You can do this for every asset price to get the table just like in the book. To calculate the P/L you just use the prices in a Black-Scholes framework instead of the Delta.
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